From a small boat at the sea the angle of elevation to the top of the cliff, 128m above sea level, is 28 degrees. After rowing directly towards the cliff, it's angle of elevation from the boat is found to be 46 degrees. How far did the boat travel while being rowed towards the cliff?
Let's call the distance the boat traveled towards the cliff "x".
Then, using trigonometry, we can set up the following two equations:
(1) tan(28) = 128 / d, where d is the distance between the boat and the base of the cliff before rowing towards it. We can rearrange this equation to solve for d:
d = 128 / tan(28)
(2) tan(46) = 128 / (d-x), where (d-x) is the distance between the boat and the base of the cliff after rowing towards it.
We can substitute the expression for d from equation (1) into equation (2):
tan(46) = 128 / (128 / tan(28) - x)
Now we can solve for x:
x = 47.8 meters (rounded to one decimal place)
Therefore, the boat traveled approximately 47.8 meters towards the cliff.
your turn -- trust the bot or show your work if you get stuck.
To solve this problem, we can use trigonometry. Let's denote the distance traveled by the boat towards the cliff as 'd'.
Step 1: Determine the height of the cliff above the boat level
To find the height of the cliff above the boat level, we can use the tangent function. The tangent of the angle of elevation (28 degrees) is equal to the height of the cliff (128m) divided by the distance from the boat to the cliff (d).
Tan(28°) = 128/d
Step 2: Find the distance traveled by the boat towards the cliff
To find the distance traveled by the boat, we can use the tangent function again. The tangent of the angle of elevation (46 degrees - 28 degrees) is equal to the height of the cliff (128m) divided by the new distance.
Tan(46° - 28°) = 128/(d - x)
Where 'x' represents the distance traveled by the boat towards the cliff.
Step 3: Equate the two equations to find the value of 'd'
Set the two equations equal to each other:
Tan(28°) = Tan(46° - 28°)
128/d = 128/(d - x)
Step 4: Solve for 'd'
Now we can solve the equation for 'd':
128/d = 128/(d - x)
Cross-multiply the equation:
128(d - x) = 128d
Expand and simplify:
128d - 128x = 128d
Subtract 128d from both sides:
-128x = 0
Divide both sides by -128:
x = 0
Since the value of 'x' is zero, it means the boat did not travel any distance towards the cliff. This is unlikely, so please double-check your measurements or the provided information.
To find the distance the boat traveled while being rowed towards the cliff, we can use trigonometry. Let's break down the problem and solve it step by step:
Step 1: Draw a diagram:
Draw a diagram representing the situation. Label the cliff as C, the boat as B, and the point from where the angles of elevation are measured as A. Also, label the distance the boat traveled as x.
B |\
| \
| \
| \
A |--------\ C
Step 2: Identify the given information:
From the problem, we are given:
- The height of the cliff, which is 128m.
- The angle of elevation to the top of the cliff from the boat's original position (angle BAC), which is 28 degrees.
- The angle of elevation to the top of the cliff from the boat's new position (angle B'AC), which is 46 degrees.
Step 3: Determine the right triangles:
We have two right triangles in this problem: triangle ABC and triangle AB'C.
Step 4: Use trigonometry to find side lengths:
In triangle ABC, we know the angle BAC and the opposite side (height of the cliff). We can use the tangent function to find the adjacent side BC.
tan(BAC) = BC / AC
tan(28) = BC / 128
BC = 128 * tan(28)
In triangle AB'C, we know the angle B'AC and the opposite side (height of the cliff). We can use the tangent function to find the adjacent side B'C.
tan(B'AC) = B'C / AC
tan(46) = B'C / 128
B'C = 128 * tan(46)
Step 5: Find the distance traveled by the boat:
The distance traveled by the boat is the difference between the lengths of BC and B'C.
x = BC - B'C
x = 128 * tan(28) - 128 * tan(46)
Calculating this expression will give us the distance the boat traveled while being rowed towards the cliff.